On the Decision Number of Graphs

Let $G$ be a graph. A good function is a function $f:V(G)\rightarrow {-1,1}$, satisfying $f(N(v))\geq 1$, for each $v\in V(G)$, where $ N(v)={u\in V(G)\, |\, uv\in E(G) } $ and $f(S) = \sum_{u\in S} f(u)$ for every $S \subseteq V(G) $. For every cubic graph $G$ of order $ n, $ we prove that $ \gamma(G) \leq \frac{5n}{7} $ and show that this inequality is sharp. A function $f:V(G)\rightarrow {-1,1}$ is called a nice function, if $f(N[v])\le1$, for each $v\in V(G)$, where $ N[v]={v} \cup N(v) $. Define $\overline{\beta}(G)=max{f(V(G))}$, where $f$ is a nice function for $G$. We show that $\overline\beta(G)\ge -\frac{3n}{7}$ for every cubic graph $G$ of order $n$, which improves the best known bound $-\frac{n}{2}$.